US8934631B2 - Decompressing apparatus and compressing apparatus - Google Patents
Decompressing apparatus and compressing apparatus Download PDFInfo
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- US8934631B2 US8934631B2 US13/225,964 US201113225964A US8934631B2 US 8934631 B2 US8934631 B2 US 8934631B2 US 201113225964 A US201113225964 A US 201113225964A US 8934631 B2 US8934631 B2 US 8934631B2
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- G—PHYSICS
- G09—EDUCATION; CRYPTOGRAPHY; DISPLAY; ADVERTISING; SEALS
- G09C—CIPHERING OR DECIPHERING APPARATUS FOR CRYPTOGRAPHIC OR OTHER PURPOSES INVOLVING THE NEED FOR SECRECY
- G09C1/00—Apparatus or methods whereby a given sequence of signs, e.g. an intelligible text, is transformed into an unintelligible sequence of signs by transposing the signs or groups of signs or by replacing them by others according to a predetermined system
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
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- G06F17/12—Simultaneous equations, e.g. systems of linear equations
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- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/30—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy
- H04L9/3066—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy involving algebraic varieties, e.g. elliptic or hyper-elliptic curves
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L2209/00—Additional information or applications relating to cryptographic mechanisms or cryptographic arrangements for secret or secure communication H04L9/00
- H04L2209/30—Compression, e.g. Merkle-Damgard construction
Definitions
- Embodiments described herein relate generally to a decompressing apparatus and a compressing apparatus for elements over a finite field.
- FIG. 1 is a block diagram of a compressing apparatus according to a first embodiment
- FIG. 2 is a flowchart of a compression processing according to the first embodiment
- FIG. 3 is a block diagram of a decompressing apparatus according to the first embodiment
- FIG. 4 is a flowchart of decompression processing according to the first embodiment
- FIG. 5 is a block diagram of a compressing apparatus according to a second embodiment
- FIG. 6 is a block diagram of a decompressing apparatus according to the second embodiment.
- FIG. 7 is a block diagram of a compressing apparatus according to a third embodiment.
- FIG. 8 is a block diagram of a decompressing apparatus according to the third embodiment.
- FIG. 9 is a block diagram of a decompressing apparatus according to a fourth embodiment.
- FIG. 10 is a hardware structure diagram of the decompressing apparatus and the compressing apparatus according to each embodiment.
- a decompressing apparatus includes an input unit, a calculating unit, a first selecting unit, and a decompressing unit.
- the input unit inputs additional data, which is obtained based on trace expression data in which an element in a subgroup of a multiplicative group of a finite field is trace-expressed and affine expression data in which the trace expression data is affine-expressed, and the trace expression data.
- the calculating unit calculates a plurality of solutions of simultaneous equations derived by the trace expression data.
- the first selecting unit selects any of a plurality of items of affine expression data in which the element is affine-expressed based on the additional data, the affine expression data being found from the solutions.
- the decompressing unit decompresses the selected affine expression data to the element.
- Fpm indicates a finite field having p ⁇ m elements, and is called an m-th degree extension field of Fp.
- F3m indicates an m-th degree extension field of a finite field F3 having three elements.
- a ⁇ b indicates the b-th power of a.
- the b-th power of a may be expressed as a b .
- a finite field Fq will be considered.
- q ⁇ 5 (mod 7) is obtained.
- An affine transform map Af:Fq6 ⁇ Fq3 and an affine transform inverse map Af ⁇ 1 ⁇ :Fq3 ⁇ Fq6 are defined in the equation (2).
- Af ( g ) ( G 1+1)/ G 2
- Af ⁇ 1 ⁇ ( Af ( g )) ( Af ( g )+ z )/( Af ( g )+ z ⁇ q ) (2)
- FIG. 1 is a block diagram illustrating an exemplary structure of a compressing apparatus 100 according to the first embodiment.
- the compressing apparatus 100 comprises an input unit 101 , a first transforming unit 102 , a second transforming unit 103 , an additional bit deciding unit 104 , an output unit 105 and a storing unit 121 .
- the input unit 101 is directed for inputting an element in an algebraic torus subset to be compressed.
- the first transforming unit 102 transforms the input element into trace expression data expressed in a trace expression. In the following, the trace expression data will be simply called trace expression.
- the first transforming unit 102 transforms the input element into the trace expression by the trace map Tr.
- the second transforming unit 103 transforms the input element into affine expression data expressed in an affine expression.
- the affine expression data will be simply called affine expression.
- the second transforming unit 103 transforms the input element into the affine expression by the affine transform map Af.
- the additional bit deciding unit 104 decides an additional bit based on the trace expression data in which the element in the subgroup of the multiplicative group of the finite field (the algebraic torus of the finite field expression) is trace-expressed and the affine expression data in which the element in the subgroup of the multiplicative group of the finite field is affine-expressed.
- the additional bit deciding unit 104 decides additional data (hereinafter referred to as additional bit below) for finding the affine expression from the solutions of predetermined simultaneous equations based on the trace expression and the affine expression.
- the additional bit is decided based on the candidates of the affine expression obtained by decompressing the trace expression data in which the element in the subgroup of the multiplicative group of the finite field (the algebraic torus of the finite field expression) is trace-expressed and the affine expression data in which the element in the subgroup of the multiplicative group of the finite field (the algebraic torus of the finite field expression) is affine-expressed.
- the output unit 105 outputs the trace expression and the additional bit.
- the storing unit 121 stores information used for finding previously-derived equations to be used for deciding the additional bit.
- FIG. 2 is a flowchart illustrating a flow of the entire compression processing according to the first embodiment.
- the input unit 101 inputs an element g in an algebraic torus subgroup (step S 101 ).
- the first transforming unit 102 inputs g into the trace map to calculate the trace expression Tr(g) (step S 102 ).
- the second transforming unit 103 inputs g into the affine map to calculate the affine expression of ( ⁇ +1, ⁇ +1, ⁇ +1) ⁇ Fq3 (step S 103 ).
- the additional bit deciding unit 104 decides multivariable simultaneous equations derived by the conditions for the trace expression and the algebraic torus and the condition for the finite field (step S 104 ).
- the additional bit deciding unit 104 solves the multivariable simultaneous equations to decide the additional bit (step S 105 ).
- the output unit 105 outputs the trace expression and the additional bit (Tr(g), i) (step S 106 ). Any method that can discriminate the six solutions may be used for deciding the additional bit not only by arranging and deciding the six solutions in descending order but also by arranging and deciding them in ascending order.
- the compression at the compression rate of 1/6 can be realized by the trace map Tr:Fq6 ⁇ Fq. i can be expressed by three bits due to 1 ⁇ i ⁇ 6. In other words, the additional bit i rarely affects the compression rate of 1/6.
- step S 104 The method for deriving the multivariable simultaneous equations (step S 104 ) and the method for solving the multivariable simultaneous equations (step S 105 ) can be achieved with the same structure as a decompressing apparatus 200 .
- the respective processing will be described in detail along with the structure of the decompressing apparatus 200 .
- FIG. 3 is a block diagram illustrating an exemplary structure of the decompressing apparatus 200 according to the first embodiment.
- the decompressing apparatus 200 comprises an input unit 201 , a calculating unit 210 , a first selecting unit 202 , a decompressing unit 203 , an output unit 204 and a storing unit 221 .
- the input unit 201 inputs the trace expression and the additional bit output from the compressing apparatus 100 .
- the calculating unit 210 derives the multivariable simultaneous equations from the input trace expression and calculates the solutions of the multivariable simultaneous equations.
- the calculating unit 210 comprises a first equation deriving unit 211 , a first solution finding unit 212 , a second equation deriving unit 213 , a second solution finding unit 214 , a third equation deriving unit 215 and a third solution finding unit 216 .
- the first equation deriving unit 211 derives the first equation obtained by inputting the input trace expression data into a preset coefficient in a previously-found k 1 -th (k 1 is a preset integer of 1 or more) degree equation over the finite field Fq.
- k 1 is a preset integer of 1 or more degree equation over the finite field Fq.
- the first solution finding unit 212 finds the solutions of the first equation.
- the third equation deriving unit 215 derives the third equation obtained by inputting at least one of the solutions found by the first solution finding unit 212 and the solutions found by the second solution finding unit 214 into a preset coefficient in a previously-found k 3 -th (k 3 is a preset integer of 1 or more) degree equation over the finite field Fq.
- k 3 is a preset integer of 1 or more degree equation over the finite field Fq.
- the third solution finding unit 216 finds the solutions of the third equation.
- the first selecting unit 202 finds a plurality of affine expressions from the plurality of solutions calculated by the calculating unit 210 and selects any affine expression corresponding to the additional data from among the found affine expressions.
- the decompressing unit 203 decompresses the selected affine expression to the element in the algebraic torus subgroup before the compression.
- the decompressing unit 203 transforms the affine expression into the pre-compression element by the affine transform inverse map Af ⁇ 1 ⁇ .
- the output unit 204 outputs the decompressed element in the algebraic torus subgroup.
- the storing unit 221 stores information for finding previously-derived equations used for deciding the additional bit as the storing unit 121 in the compressing apparatus 100 does.
- the storing unit 121 and the storing unit 221 may be configured of any generally-used storage medium such as HDD (Hard Disk Drive), optical disk, memory card or RAM (Random Access Memory).
- FIG. 4 is a flowchart illustrating a flow of the entire decompression processing in the first embodiment.
- the input unit 201 inputs the trace expression T ⁇ Fq and the additional bit j ⁇ F2 ⁇ 3 (step S 201 ).
- the calculating unit 210 derives the multivariable simultaneous equations from the input trace expression and calculates the solutions of the multivariable simultaneous equations (step S 202 ). The calculation processing by the calculating unit 210 will be detailed below.
- the first selecting unit 202 selects (decides) the affine expression based on the calculated solutions and the input additional bit (step S 203 ).
- the decompressing unit 203 decompresses the affine expression to the element g in the algebraic torus subgroup (step S 204 ).
- the output unit 204 outputs the element g in the algebraic torus subgroup which is obtained by the decompression (step S 205 ).
- step S 202 The calculation processing by the calculating unit 210 in step S 202 will be detailed below.
- the first equation deriving unit 211 derives the quadric equation A(x) expressed in the equation (3) (step S 202 ).
- a ( x ): x ⁇ 2+( T ⁇ t ⁇ 2 ⁇ +1) ⁇ 1 ⁇ 0over Fq (3)
- the first equation deriving unit 211 reads the information for finding the above A(x) from the storing unit 221 , for example.
- the information contains relationships between coefficients and variables and information for specifying a preset coefficient for inputting the trace expression.
- the first equation deriving unit 211 inputs the trace expression input into the preset coefficient to decide A(x) with reference to the information.
- the first equation deriving unit 211 sends the decided A(x) to the first solution finding unit 212 .
- the first solution finding unit 212 sends the found solutions a1 and a2 to the second equation deriving unit 213 and the third equation deriving unit 215 .
- the second equation deriving unit 213 derives the cubic equations B(x) and C(x) expressed by the equations (4) and (5).
- B ( x ): x ⁇ 3 ⁇ a 1 ⁇ x ⁇ 2 ⁇ ( a 1 ⁇ 2 ⁇ 1) x ⁇ Z ( a 1) 0 over Fq (4)
- C ( x ): x ⁇ 3 ⁇ a 2 ⁇ x ⁇ 2 ⁇ ( a 2 ⁇ 2 ⁇ 1) x ⁇ Z ( a 2) 0 over Fq (5)
- the second equation deriving unit 213 sends B(x) and C(x) to the second solution finding unit 214 .
- the second solution finding unit 214 sends the solutions (b1, b2, b3) and (c1, c2, c3) to the third equation deriving unit 215 .
- the third equation deriving unit 215 derives the linear equations D(x), E(x), F(x), G(x), H(x) and I(x) expressed in the equations (6) to (11).
- the third equation deriving unit 215 sends D(x), E(x), F(x), G(x), H(x) and I(x) to the third solution finding unit 216 .
- the third solution finding unit 216 finds the solutions d, e, f, g, h and i.
- the third solution finding unit 216 sends six combinations of solutions (a1 ⁇ b1 ⁇ d, b1, d), (a1 ⁇ b2 ⁇ e, b2, e), (a1 ⁇ b3 ⁇ f, b3, f), (a2 ⁇ c1 ⁇ g, c1, g), (a2 ⁇ c2 ⁇ h, c2, h) and (a2 ⁇ c3 ⁇ i, c3, i) from the above solutions and the solutions b1, b2, b3, c1, c2, c3 of the cubic equations B(x) and C(x) to the first selecting unit 202 .
- the first selecting unit 202 calculates (a1 ⁇ b1 ⁇ d+1, b1+1, d+1), (a1 ⁇ b2 ⁇ e+1, b2+1, e+1), (a1 ⁇ b3 ⁇ f+1, b3+1, f+1), (a2 ⁇ c1 ⁇ g+1, c1+1, g+1), (a2 ⁇ c2 ⁇ h+1, c2+1, h+1) and (a2 ⁇ c3 ⁇ i+1, c3+1, i+1), each element in the six combinations of solutions being added with 1.
- the first selecting unit 202 arranges the calculated six solutions according to a predetermined rule. The solutions are arranged in descending order in the following, but the rule is not limited to that.
- the first selecting unit 202 selects the j-th largest value from among the six arranged solutions. The value corresponds to the desired affine expression. The first selecting unit 202 sends the selected value (affine expression) to the decompressing unit 203 .
- the decompressing unit 203 transforms the received affine expression into the finite field expression of the algebraic torus subgroup by the affine transform inverse map Af ⁇ 1 ⁇ , and outputs the finite field expression.
- Tr(g′) can be calculated not by decompressing the trace expression Tr(g) but by operating the additional bit by use of the property that the trace expressions Tr(g) and Tr(g′) are identical for a certain element g over the algebraic torus subgroup and an element g′ obtained by calculating a Frobenius map.
- a decompressing apparatus uses part of additional bits to narrow equations to be solved. Thus, compression and decompression can be more efficiently performed.
- the structure of algebraic torus is the same as that of the first embodiment and thus an explanation thereof will be omitted.
- FIG. 5 is a block diagram illustrating an exemplary structure of a compressing apparatus 100 - 2 according to the second embodiment.
- the compressing apparatus 100 - 2 comprises the input unit 101 , the first transforming unit 102 , the second transforming unit 103 , an additional bit deciding unit 104 - 2 , the output unit 105 and the storing unit 121 .
- the second embodiment is different from the first embodiment in the function of the additional bit deciding unit 104 - 2 .
- Other structures and functions are the same as those of the compressing apparatus 100 according to the first embodiment illustrated in the block diagram of FIG. 1 , and thus are denoted by the same reference numerals as those in FIG. 1 and an explanation thereof will be omitted.
- the additional bit deciding unit 104 - 2 solves a quadric equation derived based on the conditions for the trace expression and the algebraic torus and the condition for the finite field, thereby finding the solutions a1 and a2.
- the additional bit deciding unit 104 - 2 derives the cubic equation based on the trace expression, the sums of elements in the affine expression, the condition for the torus, and the condition for the finite field.
- the additional bit deciding unit 104 - 2 derives the linear equation based on the solutions of the cubic equation and finds three candidates for the affine expression.
- the additional bit deciding unit 104 - 2 arranges the found candidates in descending order, compares them with the affine expression of g, and when the candidate coincides with i2-th (1 ⁇ i2 ⁇ 3) element, decides i2 as the additional bit 2.
- the output unit 105 outputs the additional bits i1 and i2 together with the trace expression.
- FIG. 6 is a block diagram illustrating an exemplary structure of the decompressing apparatus 200 - 2 according to the second embodiment.
- the decompressing apparatus 200 - 2 comprises the input unit 201 , a calculating unit 210 - 2 , a first selecting unit 202 - 2 , the decompressing unit 203 , the output unit 204 and the storing unit 221 .
- the second embodiment is different from the first embodiment in the functions of the calculating unit 210 - 2 and the first selecting unit 202 - 2 .
- the other structures and functions are the same as those of the decompressing apparatus 200 according to the first embodiment illustrated in the block diagram of FIG. 3 , and thus are denoted by the same reference numerals and an explanation thereof will be omitted.
- the calculating unit 210 - 2 further comprises a second selecting unit 217 .
- the functions of the first equation deriving unit 211 and the first solution finding unit 212 are the same as those in the first embodiment, and thus are denoted by the same reference numerals and an explanation thereof will be omitted.
- the second selecting unit 217 uses the additional bit 1 (i1) among the input additional bits to select the i1-th solution when the solutions obtained by the first solution finding unit 212 are arranged in descending order.
- the second equation deriving unit 213 - 2 uses the solution selected by the second selecting unit 217 to derive the second equation.
- the second solution finding unit 214 - 2 finds the solutions of the second equation.
- the third equation deriving unit 215 - 2 derives the third equation obtained by inputting at least one of the solution selected by the second selecting unit 217 and the solutions found by the second solution finding unit 214 - 2 .
- the third solution finding unit 216 - 2 finds the solutions of the third equation.
- the first selecting unit 202 - 2 is different from the first selecting unit 202 according to the first embodiment in that the additional data i2 among the items of additional data is used to select a solution.
- the input unit 201 sends the trace expression T to the first equation deriving unit 211 .
- the first equation deriving unit 211 derives the quadric equation A(x) by the above processing, and sends it to the first solution finding unit 212 .
- the second selecting unit 217 arranges the solutions a1 and a2 in descending order, selects the i1-th element as a and sends it to the second equation deriving unit 213 - 2 and the third equation deriving unit 215 - 2 .
- the second equation deriving unit 213 - 2 derives the cubic equation B(x) expressed in the equation (12).
- Z(x) and Y(x) are as noted above.
- B ( x ): x ⁇ 3 ⁇ a ⁇ x ⁇ 2 ⁇ ( a ⁇ 2 ⁇ 1) x ⁇ Z ( a ) 0 over Fq (12)
- the second equation deriving unit 213 - 2 sends B(x) to the second solution finding unit 214 - 2 .
- the second solution finding unit 214 - 2 sends the solution (b1, b2, b3) to the third equation deriving unit 215 - 2 .
- the third equation deriving unit 215 - 2 derives the linear equations D(x), E(x) and F(x) expressed in the equations (13) to (15).
- D ( x ):( ⁇ b 1 ⁇ 2 ⁇ +( a +(1 /a )) ⁇ b 1) ⁇ x+Z ( a )+( ⁇ a +(1 /a )) ⁇ b 1 ⁇ 2 ⁇ ( Y ( a )/ a ) ⁇ b 1 0 over Fq (13)
- E ( x ):( ⁇ b 2 ⁇ 2 ⁇ +( a +(1 /a )) ⁇ b 2) ⁇ x+Z ( a )+( ⁇ a +(1 /a )) ⁇ b 2 ⁇ 2 ⁇ ( Y ( a )/ a ) ⁇ b 2 0 over Fq (14)
- the third equation deriving unit 215 - 2 sends the linear equations D(x), E(x) and F(x) to the third solution finding unit 216 - 2 .
- the third solution finding unit 216 - 2 finds their solutions d, e and f.
- the third solution finding unit 216 - 2 sends three combinations of solutions (a ⁇ b1 ⁇ d, b1, d), (a ⁇ b2 ⁇ e, b2, e) and (a ⁇ b3 ⁇ f, b3, f) to the first selecting unit 202 - 2 .
- the first selecting unit 202 - 2 calculates (a ⁇ b1 ⁇ d+1, b1+1, d+1), (a ⁇ b2 ⁇ e+1, b2+1, e+1) and (a ⁇ b3 ⁇ f+1, b3+1, f+1), each element in the three combinations of solutions being added with 1.
- the first selecting unit 202 - 2 arranges the three calculated solutions in descending order.
- the first selecting unit 202 - 2 selects the i2-th largest solution among the three arranged solutions.
- the value corresponds to the desired affine expression.
- the first selecting unit 202 - 2 sends the selected value (affine expression) to the decompressing unit 203 .
- the decompressing unit 203 transforms the received affine expression into the finite field expression of the algebraic torus subgroup and outputs the resulting finite field expression.
- the compressing apparatus and the decompressing apparatus can separately calculate the additional bits i1 and i2 and can use the additional bit i1 to early narrow the equations to be solved. Thereby, the compression and the decompression can be more efficiently performed.
- the affine transform map Af:Fq4 ⁇ Fq2 and the affine transform inverse map Af ⁇ 1 ⁇ :Fq2 ⁇ Fq4 are defined in the equation (17).
- Af ( g ) ( G+ 1)/ G 2 ⁇ Fq 2
- Af ⁇ 1 ⁇ ( Af ( g )) ( Af ( g )+ z )/( Af ( g )+ z ⁇ q ) ⁇ Fq 4 (17)
- FIG. 7 is a block diagram illustrating an exemplary structure of a compressing apparatus 100 - 3 according to a third embodiment.
- the compressing apparatus 100 - 3 comprises the input unit 101 , a first transforming unit 102 - 3 , a second transforming unit 103 - 3 , an additional bit deciding unit 104 - 3 , the output unit 105 and the storing unit 121 .
- the constituents having the same functions as those of the first embodiment are denoted by the same reference numerals as those in FIG. 1 and an explanation thereof will be omitted.
- the first transforming unit 102 - 3 transforms the element g input from the trace map Tr in the equation (16) into the trace expression Tr(g).
- the second transforming unit 103 - 3 transforms the element g input from the affine transform map Af in the equation (17) into the affine expression ( ⁇ , ⁇ ) ⁇ Fq ⁇ Fq.
- the additional bit deciding unit 104 - 2 decides the least significant bit of ⁇ as the additional bit 1 and the least significant bit of ⁇ as the additional bit 2.
- FIG. 8 is a block diagram illustrating an exemplary structure of a decompressing apparatus 200 - 3 according to the third embodiment.
- the decompressing apparatus 200 - 3 comprises the input unit 201 , a calculating unit 210 - 3 , a first selecting unit 202 - 3 , a decompressing unit 203 - 3 , the output unit 204 and the storing unit 221 .
- the constituents having the same functions as those of the first embodiment are denoted by the same reference numerals as those in FIG. 3 and an explanation thereof will be omitted.
- the calculating unit 210 - 3 comprises a first equation deriving unit 211 - 3 , a first solution finding unit 212 - 3 , a second equation deriving unit 213 - 3 and a second solution finding unit 214 - 3 .
- the first equation deriving unit 211 - 3 derives the first equation as a different quadric equation from the first embodiment.
- the first solution finding unit 212 - 3 finds the solutions of the first equation.
- the second equation deriving unit 213 - 3 derives the second equation as a quadric equation, unlike the first embodiment.
- the second solution finding unit 214 - 3 finds the solutions of the second equation.
- the first selecting unit 202 - 3 selects any solution corresponding to the additional data from among the solutions calculated by the calculating unit 210 - 3 .
- the first equation deriving unit 211 - 3 derives the quadric equation A(x) expressed in the equation (18) and sends it to the first solution finding unit 212 - 3 .
- a ( x ): x ⁇ 2 +x +1 ⁇ T ⁇ q ⁇ t ⁇ 0 over Fq (18)
- the first solution finding unit 212 - 3 may transform the term of x ⁇ 2 into a linear term by using a Frobenius map, thereby solving the transformed linear equation.
- the second equation deriving unit 213 - 3 derives the quadric equations B(x) and C(x) expressed in the equations (19) and (20) and sends them to the second solution finding unit 214 - 3 .
- B ( x ): x ⁇ 2 +x+a 1 ⁇ 2 +a 1 ⁇ t ⁇ T ⁇ q ⁇ t ⁇ 0 over Fq (19)
- C ( x ): x ⁇ 2 +x+a 2 ⁇ 2 +a 2 ⁇ t ⁇ T ⁇ q ⁇ t ⁇ 0 over Fq (20)
- the second solution finding unit 214 - 3 sends the solutions b1, b2, c1 and c2 to the first selecting unit 202 - 3 .
- the first selecting unit 202 - 3 finds four combinations of solutions (a1 ⁇ b1, b1), (a1 ⁇ b2, b2), (a2 ⁇ c1, c1) and (a2 ⁇ c2, c2).
- the first selecting unit 202 - 3 compares the least significant bit of the first component in each combination (a1 ⁇ b1, a1 ⁇ b2, a2 ⁇ c1, a2 ⁇ c2) with the additional bit 1, compares the least significant bit of the second component in each combination (b1, b2, c1, c2) with the additional bit 2, and selects the coincident pair.
- the value of the pair corresponds to the desired affine expression.
- the first selecting unit 202 - 3 sends the selected value (affine expression) to the decompressing unit 203 - 3 .
- the decompressing unit 203 - 3 transforms the received affine expression into the finite filed expression of the algebraic torus subgroup and outputs it.
- the third embodiment can realize the compression at the compression rate of 1/4 and the decompression similar to the first embodiment.
- a compressing apparatus according to the fourth embodiment is the same as the compressing apparatus 100 - 3 according to the third embodiment.
- FIG. 9 is a block diagram illustrating an exemplary structure of a decompressing apparatus 200 - 4 according to the fourth embodiment.
- the decompressing apparatus 200 - 4 comprises the input unit 201 , a calculating unit 210 - 4 , a first selecting unit 202 - 4 , a decompressing unit 203 - 3 , the output unit 204 and the storing unit 221 .
- the constituents having the same functions as those of the third embodiment are denoted by the same reference numerals as those in FIG. 8 and an explanation thereof will be omitted.
- the calculating unit 210 - 4 comprises a first equation deriving unit 211 - 3 , the first solution finding unit 212 - 3 , a second selecting unit 217 - 4 , a second equation deriving unit 213 - 4 and a second solution finding unit 214 - 4 .
- the second selecting unit 217 - 4 uses the additional bit 1(i1) and the additional bit 2(i2) among the input additional bits to compare the least significant bit of the solutions obtained by the first solution finding unit 212 - 3 with the sum of the additional bit 1(i1) and the additional bit 2(i2) over F2, thereby selecting the coincident solution.
- the second equation deriving unit 213 - 4 uses the solution selected by the second selecting unit 217 - 4 to decide the second equation.
- the second solution finding unit 214 - 4 finds the solutions of the second equation.
- the first equation deriving unit 211 - 3 derives the quadric equation A(x) expressed in the equation (18) and sends it to the first solution finding unit 212 - 3 .
- the first solution finding unit 212 - 3 sends the solutions a1 and a2 to the second selecting unit 217 - 4 .
- a Frobenius map may be used to transform the term of x ⁇ 2 into a linear term to solve the transformed linear equation.
- the second selecting unit 217 - 4 compares the sum of the additional bit 1(i1) and the additional bit 2(i2) over F2 with the least significant bit of the solutions a1, a2, and selects the coincident solution as the solution a.
- the second selecting unit 217 - 4 sends the selected solution a to the second equation deriving unit 213 - 4 .
- the second equation deriving unit 213 - 4 derives the quadric equation B(x) expressed in the equation (21) and sends it to the second solution finding unit 214 - 4 .
- B ( x ): x ⁇ 2 +x+a ⁇ 2 +a ⁇ t ⁇ T ⁇ q ⁇ t ⁇ 0 over Fq (21)
- the second solution finding unit 214 - 4 sends the solutions b1 and b2 to the first selecting unit 202 - 4 .
- the first selecting unit 202 - 4 compares the additional bit 2(i2) with the least significant bit of the solutions b1, b2, and selects the coincident solution as the solution b.
- the first selecting unit 202 - 4 sends (a ⁇ b, b) ⁇ Fq ⁇ Fq obtained from the solutions a and b to the decompressing unit 203 - 3 .
- (a ⁇ b, b) corresponds to the desired affine expression.
- the decompressing unit 203 - 3 transforms the received affine expression into the finite field expression of the algebraic torus subgroup and outputs it.
- the fourth embodiment can realize the compression at the compression rate of 1/4 and the decompression similar to the second embodiment.
- the compressing apparatus and the decompressing apparatus according to each of the above embodiments can be provided inside an apparatus for encoding and decoding a public key encryption, for example.
- the compressing apparatus may be provided in an information processing apparatus (encoding apparatus) for transmitting the data encoded by the public key encryption and the decompressing apparatus may be provided in an information processing apparatus (decoding apparatus) for receiving and decoding the encoded data.
- a compressing/decompressing apparatus including both the compressing apparatus and the decompressing apparatus according to each of the above embodiments may be configured.
- T6(Fq) is assumed as torus and the torus subgroup takes a trace expression for compressing the expression.
- the subgroups can be compatibly configured by the above factorization.
- the subgroup having the order (q+ ⁇ (3q)+1) is assumed as G+ and the subgroup having the order (q ⁇ (3q)+1) is assumed as G_. Since the mapping from the element in the subgroup of T6(Fq) into the element in the trace expression is not bijective, a plurality of elements for T6(Fq) are present for an inverse map of a certain element in the trace expression. Since six elements are present for T6(Fq) corresponding to one trace expression, at least three additional bits are required for identifying the six elements. Since the bit length required for the expression is log — 2(q)+3 bits including the three additional bits for both G+ and G_, G_ as a smaller group than q is not necessarily required but G_ is taken as a subgroup here.
- the extension field Fq6 obtained by six-order decompressing the finite field Fq by the modulus polynomial ⁇ 7 is assumed and the compression/decompression map for the elements in the subgroup G_ of T6(Fq) is configured.
- ⁇ 7 1 +x+x q 4 +x q 5 +x q 2 +x q +x q 3 (23)
- Tr ( g ) g+g q +g q 2 +g q 3 +g q 4 +g q 5 (29)
- 1 is a unit element of the multiplication of Fq6.
- ⁇ ⁇ ( g ) - ( ⁇ 0 + 1 ) ⁇ 1 ⁇ F q 3 ( 30 )
- the equation (29) corresponds to the trace map Tr in the equation (1).
- the equations (30) and (31) correspond to the affine transform map Af and the affine transform inverse map Af ⁇ 1 ⁇ in the equation (2).
- Tr(g) is expressed by the sum of elements in h ⁇ 1 .
- h is expressed in the equation (35).
- Tr(g) is obtained.
- the denominator of Tr(g) is expressed in the equation (36). Since the numerator will be described later and is omitted here.
- Tr ⁇ ( g ) ⁇ denominator ] ( ⁇ 1 6 + ⁇ 2 6 + ⁇ 3 6 ) + ( ⁇ 1 ⁇ ⁇ 2 5 + ⁇ 2 ⁇ ⁇ 3 5 + ⁇ 3 ⁇ ⁇ 1 5 ) + ( ⁇ 1 4 ⁇ ⁇ 2 2 + ⁇ 2 4 ⁇ ⁇ 3 2 + ⁇ 2 4 ⁇ ⁇ 1 2 ) + ( ⁇ 1 4 ⁇ ⁇ 2 ⁇ ⁇ 3 + ⁇ 1 ⁇ ⁇ 2 4 ⁇ ⁇ 3 + ⁇ 1 ⁇ ⁇ 2 ⁇ ⁇ 3 4 ) + ( ⁇ 1 2 ⁇ ⁇ 2 4 + ⁇ 2 2 ⁇ ⁇ 3 4 + ⁇ 3 2 ⁇ ⁇ 1 4 ) - ( ⁇ 1 3 ⁇ ⁇ 2 3 + ⁇ 2 3 ⁇ ⁇ 1 3 ) - ( ⁇ 1 3 ⁇ ⁇ 2 3 + ⁇ 2 3 ⁇ ⁇ 1 3 ) - ( ⁇ 1 3 ⁇ ⁇ 2 3 + ⁇
- the conditional equation for ⁇ 1, ⁇ 2, and ⁇ 3 is arranged from the above equation. It is expressed in the equation (44).
- the trace value of an element is the same as the trace value of the q-th power of the element. From the property, at least six affine expressions having the same trace value are present. The relationship between the affine expressions will be considered. In the case of f ⁇ Fq3, the relational equation is expressed in the equation (47).
- Tr ⁇ ( g ) ⁇ denominator ] ( ⁇ + ⁇ + ⁇ ) 6 + ( ⁇ ⁇ ⁇ ⁇ 5 + ⁇ ⁇ ⁇ ⁇ 5 + ⁇ ⁇ ⁇ ⁇ 5 ) + ( ⁇ 4 ⁇ ⁇ 2 + ⁇ 4 ⁇ ⁇ 2 + ⁇ 4 ⁇ ⁇ 2 ) + ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ( ⁇ + ⁇ + ⁇ ) 3 + ( ⁇ 2 ⁇ ⁇ 4 + ⁇ 2 ⁇ ⁇ 4 + ⁇ 2 ⁇ ⁇ 4 ) - ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ( ⁇ 2 ⁇ ⁇ + ⁇ 2 ⁇ ⁇ + ⁇ 2 ⁇ ⁇ ) - ⁇ 2 ⁇ ⁇ 2 ⁇ ⁇ 2 - ( ⁇ 3 ⁇ ⁇ + ⁇ 3 ⁇ ⁇ ) - ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇
- Equation (51) and (52) are transformed by the equation (53).
- the two kinds of finally symmetrical equations ( ⁇ + ⁇ + ⁇ ) and ⁇ , and an asymmetrical equation ( ⁇ 2 + ⁇ 2 + ⁇ 2 ) are used as many times as possible.
- the equation (59) is obtained from the equations (57) and (58).
- Tr(g) is deformed to obtain the equations (63) and (64) from the equations (51) and (52).
- Tr ⁇ ( g ) ⁇ denominator ] ⁇ ( ⁇ + ⁇ + ⁇ ) 6 + ( ⁇ ⁇ ⁇ ⁇ 5 + ⁇ ⁇ ⁇ ⁇ 5 + ⁇ ⁇ ⁇ ⁇ 5 ) + ⁇ ( ⁇ 4 ⁇ ⁇ 2 + ⁇ 4 ⁇ ⁇ 2 ) + ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ( ⁇ + ⁇ + ⁇ ) 3 + ⁇ ( ⁇ 2 ⁇ ⁇ 4 + ⁇ 2 ⁇ ⁇ 4 + ⁇ 2 ⁇ ⁇ 4 ) - ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ( ⁇ 2 ⁇ ⁇ + ⁇ 2 ⁇ ⁇ + ⁇ 2 ⁇ ⁇ ) + ⁇ 2 ⁇ ⁇ 2 ⁇ ⁇ 2 ⁇ ) + ⁇ 2 ⁇ ⁇ 2 ⁇ ⁇ 2 ⁇ ) + ⁇ 2 ⁇ ⁇ 2 ⁇ ⁇ 2 ⁇ - ⁇ ( ⁇ 3 ⁇ +
- the equation (68) is obtained from the equation (65).
- Equation (75) is substituted into the equations (72), (73) and (74) for rearrangement to obtain the equations (76) to (78).
- the subgroup having the order of (q+ ⁇ (2q)+1) is G+ and the subgroup having the order of (q ⁇ (2q)+1) is G_. Since the mapping from an element in the subgroup in the torus into an element in the trace is not bijective, there are multiple elements of the torus to be mapped from elements of a trace. Since four elements in a trace corresponding to the elements in one trace are present (see the trace expression for details), two additional bits are required for identifying the four elements. In the following, a relationship between the trace and the affine expression will be made clear and the method for decompressing the trace to the affine expression will be described. The following operations are over F(2 ⁇ n) ⁇ 4 unless otherwise noted.
- the equation (92) collectively describes the relational equations of f used for deforming the equations in the following.
- Tr(g) g+g q +g ⁇ q 2 +g ⁇ q 3 (corresponding to the equation (16)).
- the equation (93) indicates the calculation process.
- t ⁇ q (mod 5) ⁇ 2 (mod 5) is assumed.
- the deformed equation expressed in the equation (95) is possible from the equation (94).
- f q + 1 ⁇ ⁇ 1 2 + ⁇ 1 ⁇ ⁇ 2 + ⁇ 2 2 ⁇ ( described ⁇ ⁇ above )
- Tr(g) for T4
- Tr(g ⁇ q 2 ) g ⁇ q 2 +g ⁇ q 3 +g+g q
- the trace of the element which is obtained by raising the element g of a torus to the q-th power has the same value of the original trace.
- Equation (102) is substituted into the left side of the equation (103) to obtain the equation (104).
- Tr ⁇ ( g ) ⁇ 1 2 + ⁇ 2 2 + ⁇ 1 + ⁇ 2 + 1 ⁇ 1 4 + ⁇ 1 2 ⁇ ⁇ 2 2 + ⁇ 2 4 + ⁇ 1 3 + ⁇ 2 3 + ⁇ 1 ⁇ ⁇ 2 + ⁇ 2 2 + ⁇ 2 + 1 ( 109 )
- the four solutions are in the q-th power symmetric relationship.
- the relationship of ( ⁇ 1 , ⁇ 2 ) ⁇ ( ⁇ 2 +1, ⁇ 1 ) ⁇ ( ⁇ 1 +1, ⁇ 2 +1) ⁇ ( ⁇ 2 +1, ⁇ 1 )( ⁇ ( ⁇ 1 , ⁇ 2 )) is obtained.
- the arrow goes to the right each time the q-th power is raised and returns to the origin with the q 4 -th power.
- Tr(g) is found from the equation (109).
- b 0 is assumed as the least significant bit of ⁇ 1 and b 1 is assumed as the least significant bit of ⁇ 2 to output Tr(g), (b 0 , b 1 ).
- the specific method for recovering the element of the T4 torus from the element of the trace by adding the additional bit has been described above.
- the expression can be compressed only to 1 ⁇ 3 relative to the size of the finite field in the conventional torus, but the expression can be gradually compressed to 1 ⁇ 4 with the embodiments.
- the compressing apparatus and the decompressing apparatus can be efficiently realized for the modulus polynomials which cannot efficiently configure the decompressed map in the conventional method.
- FIG. 10 is an explanatory diagram illustrating a hardware structure of the decompressing apparatus and the compressing apparatus according to the first to fourth embodiments.
- the decompressing apparatus and the compressing apparatus comprise a control device such as a CPU (Central Processing Unit) 51 , storage devices such as ROM (Read Only Memory) 52 and RAM (Random Access Memory) 53 , a communication I/F 54 connected to a network for making communication, and a bus 61 for interconnecting the respective units.
- a control device such as a CPU (Central Processing Unit) 51
- storage devices such as ROM (Read Only Memory) 52 and RAM (Random Access Memory) 53
- ROM Read Only Memory
- RAM Random Access Memory
- the programs to be executed by the decompressing apparatus and the compressing apparatus according to the first to fourth embodiments are previously incorporated in the ROM 52 or the like to be provided.
- the programs to be executed by the decompressing apparatus and the compressing apparatus according to the first to fourth embodiments may be recorded in a computer-readable recording medium such as CD-ROM (Compact Disk Read Only Memory), flexible disk (FD), CD-R (Compact Disk Recordable) or DVD (Digital Versatile Disk) in an installable form or executable form to be provided as a computer program product.
- a computer-readable recording medium such as CD-ROM (Compact Disk Read Only Memory), flexible disk (FD), CD-R (Compact Disk Recordable) or DVD (Digital Versatile Disk) in an installable form or executable form to be provided as a computer program product.
- the programs to be executed by the decompressing apparatus and the compressing apparatus according to the first to fourth embodiments may be stored on a computer connected to a network such as the Internet and may be downloaded via the network to be provided.
- the programs to be executed by the decompressing apparatus and the compressing apparatus according to the first to fourth embodiments may be provided or distributed via the network such as Internet.
- the programs to be executed by the decompressing apparatus and the compressing apparatus according to the first to fourth embodiments can cause a computer to function as each unit in the decompressing apparatus and the compressing apparatus.
- the computer may read and execute a program from a computer-readable recording medium on a min storage device.
Abstract
Description
Tr(g)=g+g^q+g^{q^2}+g^{q^3}+g^{q^4}+g^{q^5} (1)
Af(g)=(G1+1)/G2,
Af^{−1}(Af(g))=(Af(g)+z)/(Af(g)+z^q) (2)
A(x):x^2+(T^{t−2}+1)^{−1}=0over Fq (3)
B(x):x^3−a1×x^2−(a1^2−1)x−Z(a1)=0 over Fq (4)
C(x):x^3−a2×x^2−(a2^2−1)x−Z(a2)=0 over Fq (5)
D(x):(−b1^{2}+(a1+(1/a1))×b1)×x+Z(a1)+(−a1+(1/a1))×b1^2−(Y(a1)/a1)×b1=0 over Fq (6)
E(x):(−b2^{2}+(a1+(1/a1))×b2)×x+Z(a1)+(−a1+(1/a1))×b2^2−(Y(a1)/a1)×b2=0 over Fq (7)
F(x):(−b3^{2}+(a1+(1/a1))×b3)×x+Z(a1)+(−a1+(1/a1))×b3^2−(Y(a1)/a1)×b3=0 over Fq (8)
G(x):(−c1^{2}+(a2+(1/a2))×c1)×x+Z(a2)+(−a2+(1/a2))×c1^2−(Y(a2)/a2)×c1=0 over Fq (9)
H(x):(−c2^{2}+(a2+(1/a2))×c2)×x+Z(a2)+(−a2+(1/a2))×c2^2−(Y(a2)/a2)×c2=0 over Fq (10)
I(x):(−c3^{2}+(a2+(1/a2))×c3)×x+Z(a2)+(−a2+(1/a2))×c3^2−(Y(a2)/a2)×c3=0 over Fq (11)
B(x):x^3−a×x^2−(a^2−1)x−Z(a)=0 over Fq (12)
D(x):(−b1^{2}+(a+(1/a))×b1)×x+Z(a)+(−a+(1/a))×b1^2−(Y(a)/a)×b1=0 over Fq (13)
E(x):(−b2^{2}+(a+(1/a))×b2)×x+Z(a)+(−a+(1/a))×b2^2−(Y(a)/a)×b2=0 over Fq (14)
F(x):(−b3^{2}+(a+(1/a))×b3)×x+Z(a)+(−a+(1/a))×b3^2−(Y(a)/a)×b3=0 over Fq (15)
Tr(g)=g+g^q+g^{q^2}+g^{q^3} (16)
Af(g)=(G+1)/G2εFq2,
Af^{−1}(Af(g))=(Af(g)+z)/(Af(g)+z^q)εFq4 (17)
A(x):x^2+x+1−T^{q−t}=0 over Fq (18)
B(x):x^2+x+a1^2+a1^t×T^{q−t}=0 over Fq (19)
C(x):x^2+x+a2^2+a2^t×T^{q−t}=0 over Fq (20)
B(x):x^2+x+a^2+a^t×T^{q−t}=0 over Fq (21)
Φ7=1+x+x 2 +x 3 +x 4 +x 5 +x 6 ,x 7=1,
q=5 mod 7,
q=3n,(nεZ),
(n=5 mod 12) (22)
Φ7=1+x+x q
y=x+x q
z=x+x q
f=δ 1 y+δ 2 y q+δ3 y q
f t=δ1 t y+δ 2 t y q+δ3 y q
(2.2) Trace Map and Affine Map
Tr(g)=g+g q +g q
ψ−1(ψ(g))=gεT 6(F q)⊂F q
(4) Conditional Equation of G_in T6(Fq)
z:f q+t +f q+1 −f t+1 −f q+2=0,
z q :f q+t +f q+1 −f t+1 −f q+2=0 (40)
y:δ 1 t(δ3−δ2)+δ3 t(δ1+δ2+δ3)+2δ3δ1+δ1δ2+δ1 2+δ2 2−δ3=2, (41)
y q:δ2 t(δ1−δ3)+δ1 t(δ1+δ2+δ3)+2δ1δ2+δ2δ3+δ2 2+δ3 2−δ1=2, (42)
y q
(4.1) Conditional Equation of T2(Fq)
f q
f q
→{(f q+1)q
→{(δ1 2+δ2 2+2δ1δ3+δ1δ2)+(δ2 2+δ3 2+2δ1δ2+δ2δ3)+(δ3 2+δ1 2+2δ2δ3+δ1δ3)}(y+y q +y q
→{(2δ1 2+2δ2 2+2δ3 2)−(δ1+δ2+δ3)}(y+y q +y q
→{(δ1 2+δ2 2+δ3 2)+(δ1+δ2+δ3)}−1=0,
→{(δ1 2+δ2 2+δ3 2)+(δ1+δ2+δ3)}=1 (46)
(5) Affine Expression Having the Same Trace Value
-
- (δ1, δ2, δ3)
- (2δ3−1, 2δ1−1, 2δ2−1)
- (δ2, δ3, δ1)
- (2δ1−1, 2δ2−1, 2δ3−1)
- (δ3, δ1, δ2)
- (2δ2−1, 2δ3−1, 2δ1−1)
f=(α+1)y+(β+1)y q+(γ+1)y q
(5.2) Conditional Equation of T2(Fq) (Version 2)
α2+β2+γ2=1 (53)
(5.3) Conditional Equation of G_ in T6(Fq) (Version 2)
αt(γ−β)+γt(α+β+γ)−αγ+αβ+α2+β2=1
βt(α−γ)+αt(α+β+γ)−βα+βγ+β2+γ2=1
γt(β−α)+βt(α+β+γ)−γβ+γα+γ2+α2=1
(5.4) Calculation of T6(Fq) Trace (Version 3)
(α30 β+γ)6−{(α+β+γ)−(α+β+γ)3}(αβ2+βγ2+γα2)+(αβ2+βγ2+γα2)2,
=−1−(α+β+γ)6−αβγ(α+β+γ)3,
→(αβ2+βγ2+γα2)2=(α+β+γ)6+{(α+β+γ)−(α+β+γ)3}(αβ2+βγ2+γα2)−αβγ(α+β+γ)3−1
→B 2 =A 6 −A 3(B+C)+AB−1 (60)
(6) Transform from Trace into Torus
(7) Method for Deriving Cubic Equation
A=α+β+γ (71)
B=αβ 2+βγ2+γα2 (72)
C=αβγ (73)
α2+β2+γ2=1 (74)
α=A−β−γ (75)
−β3+γ3 +Aβ 2 +Aβγ+Aγ 2 +A 2 γ−B=0 (76)
−β2γ−βγ2 +Aβγ−C=0 (77)
−β2−βγ−γ2 +Aβ+Aγ+A 2−1=0 (78)
−β3−β2γ−βγ2 +Aβ 2 +Aβγ+(A 2−1)β=0 (79)
β3 −Aβ 2−(A 2−1)β−C=0 (80)
x 3 −Ax 2−(A 2−1)x−C=0
(8) Method for Deriving Linear Equation
−β2γ−βγ2−γ3 +Aβγ+Aγ 2+(A 2−1)γ=0 (81)
−γ3 +Aγ 2+(A 2−1)γ+C=0 (82)
Aβγ−Aγ 2+(−A 2+1)β+(−A 2−1)γ−B=0 (83)
Aβ 2 γ−Aβγ 2+(−A 2+1)β2+(−A 2−1)βγ−Bβ=0 (84)
−Aβ 2 γ−Aβγ 2 +A 2 βγ−AC=0 (85)
−Aβ 2γ+(−A 2+1)β2+(A 2−1)βγ−Bβ+AC=0 (86)
(−β2+(A−(1/A))β)γ+(−A+(1/A)β2 −Bβ/A+C=0 (87)
(−β2+(A−(1/A))β)x+(−A+(1/A))β2 −Bβ/A+C=0 (88)
x 2 =−[{Tr(g)}t−2+1]−1
−x 3 +Ax 2+(A 2−1)x+C=0
(−Aβ 2+(A 2−1)β)x+(−A 2+1)β2 −Bβ+AC=0 (89)
-
- To obtain two solutions by solving x2=−[{Tr(g)}t−2+1]−1 which is a quadric equation having the root of α+β+γ.
- To obtain three solutions by solving x3−Ax2−(A2−1)x−C=0 which is a cubic equation having the root of β found by substituting the two solutions into A, respectively.
- To obtain one solution and calculate a corresponding α by solving (−β2+(A−(1/A))βx+(−A+(1/A))β2−Bβ/A+C=0 which is a linear equation having the root of γ, the linear equation being obtained by substituting one of the solutions of the quadric equation into A and one of the solutions of the cubic equation corresponding to A into β.
(2.2) Frequently-used Relational Equations
(3) Trace Expressed in Components of Torus
(4) Condition for G—
→f q+t(z+z q
→f q+t +f q+1 +f t+1 +f q y+f t y q+(1+x q+1+x q
→f q+t +f q+1 +f t+1 +f q y+f t y q +y q=0 (95)
→f q+t +f q+1 +f t+1 +f q y+f t y q +y q=0,
→(δ1 t+1 y+δ 2 t+1 y q+δ1 tδ2+δ1δ2 t)+(δ1 2+δ1δ2+δ2 2)+δ1 t+1+δ2 t+1+δ1 tδ2 y+δ 1δ2 t y q)+(δ1+δ2 y q)+(δ1 t y+δ 2 t)+y q=0,
→(δ2 t+1+δ1δ2 t+δ1 t)y+(δ1 t+1+δ1 tδ2+δ2+1)y q+(δ1 2+δ1δ2+δ2 2+δ1+δ2 t)=0 (97)
y:δ 2 t+1+δ1δ2 t+δ1 t+(δ1 2+δ1δ2+δ2 2+δ1+δ2 t)=0,
y q:δ1 t+1+δ1 tδ2+δ2+1+(δ1 2+δ1δ2+δ2 2+δ1+δ2 t)=0 (98)
δ1 t+1+δ1δ2 t+δ1 t+δ2 t+1+δ1 tδ2+δ2+1=0,
→δ1 t+1+δ2 t+1+δ1 tδ2+δ1δ2 t+δ1 t+δ2+1=0,
→(δ1+δ2)t+1+δ1 t+δ2+1=0. (99)
(δ1+δ2)=(δ1 t+δ2+1)t−1,
→(δ1+δ2)(δ1 t+δ2+1)=(δ1 t+δ2+1)t,
→(δ1+δ2)(δ1 t+δ2+1)=δ1 2+δ2 t+1 (101)
(δ1+δ2+1)(δ1+δ2 t)=δ1 t+δ2 2 (102)
(δ1+δ2)(δ1 t+δ2+1)=δ1 2+δ2 t+1,
→δ1+δ2 t=δ1 t(δ1+δ2)+δ1δ2+δ2 2+δ1+δ2+δ1 2+δ1+1,
→δ1+δ2 t=δ1 t(δ1+δ2)+δ1δ2+δ2 2+δ2+δ1 2+1,
→(δ1+δ2 t)(δ1+δ2+1)={δ1 t(δ1+δ2)+δ1δ2+δ2 2+δ2+δ1 2+1}(δ1+δ2+1),
→(δ1+δ2 t)(δ1+δ2+1)=δ1 t(δ1+δ2)2+δ1 t(δ1+δ2)+(δ1+δ2+1)2δ2+(δ1 2+1)(δ1+δ2+1),
→(δ1+δ2 t)(δ1+δ2+1)=δ1 t(δ1 2+δ2 2+δ1+δ2)+δ1 3+δ2 3+δ1 2+δ1+1=0 (103)
→δ1 t+δ2 2=δ1 t(δ1 2+δ2 2+δ1+δ2)+δ1 3+δ2 3+δ1 2+δ1+1,
→δ1 t(δ1 2+δ2 2+δ1+δ2+1)+δ1 3+δ2 3+δ1 2+δ2 2+δ1+1=0 (104)
δ1 t(δ1 2+δ2 2+δ1+δ2+1)+δ1 3+δ2 3+δ1 2+δ2 2+δ1+1=0,
→(δ2 t+1)(δ1 2+δ2 2+δ1+δ2+1)+(δ2+1)3+δ1 3+(δ2+1)2+δ1 2+(δ2+1)+1=0,
→δ2 t(δ1 2+δ2 2+δ1+δ2+1)+(δ1 2+δ2 2+δ1+δ2+1)+(δ2 3+δ2 2+δ2+1)+δ1 3+(δ2 2+1)+δ1 2+(δ2+1)+1=0,
→δ2 t(δ1 2+δ2 2+δ1+δ2+1)+δ1 3+δ2 3+δ2 2+δ1+δ2+1=0 (105)
(δ1 t+δ2 t)(δ1 2+δ2 2+δ1+δ2+1)+δ1 2+δ2=0 (106)
δ1 2+δ2=(δ1 t+δ2 t)(δ1 2+δ2 2+δ1+δ2+1) (107)
δ1+δ2 2+1=(δ1 t+δ2 t+1)(δ1 2+δ2 2+δ1+δ2+1),
δ1+δ2 2=(δ1 t+δ2 t+1)(δ1 2+δ2 2+δ1+δ2+1)+1 (108)
(6.2) Deformation of Trace Equation
(δ1 4+δ1 2δ2 2+δ2 4+δ1 3+δ2 3+δ1δ2+δ2 2+δ2+1){Tr(g)}=(δ1 2+δ2 2+δ1+δ2+1),
→δ1 4+δ1 2δ2 2+δ2 4+δ1 3+δ2 3+δ1δ2+δ2 2+δ2+1=(δ1 2+δ2 2+δ1+δ2+1){Tr(g)}−1 (110)
δ1 4+δ1 2δ2 2+δ2 4+δ1 3+δ2 3+δ1δ2+δ2 2+δ2+1=(δ1 2+δ2 2+δ1+δ2+1){Tr(g)}−1,
→(δ1 4+δ2 4)+(δ2 2+δ1)(δ1 2+δ2)+(δ2 2+δ1 2)+(δ1 2+δ2)+1=(δ1 2+δ2 2+δ1+δ2+1){Tr(g)}−1,
→(δ1+δ2)4+(δ1+δ2)2+1+(δ1 2+δ2)(δ2 2+δ1+1)=(δ1 2+δ2 2+δ1+δ2+1){Tr(g)}−1 (111)
(δ1+δ2)4+(δ1+δ2)2+1+(δ1 2+δ2)(δ2 2+δ1+1)=(δ1 2+δ2 2+δ1+δ2+1){Tr(g)}−1,
→(δ1+δ2)4+(δ1+δ2)2+1+(δ1 t+δ2 t)(δ1 t+δ2 t+1)(δ1 2+δ2 2+δ1+δ2+1)2=(δ1 2+δ2 2+δ1+δ2+1){Tr(g)}−1,
→(δ1 2+δ2 2+δ1+δ2+1)2+(δ1 t+δ2 t)(δ1 t+δ2 t+1)(δ1 2+δ2 2+δ1+δ2+1)2=(δ1 2+δ2 2+δ1+δ2+1){Tr(g)}−1,
→(δ1 2+δ2 2+δ1+δ2+1)+(δ1 t+δ2 t)(δ1 t+δ2 t+1)(δ1 2+δ2 2+δ1+δ2+1)={Tr(g)}−1,
→(δ1 2+δ2 2+δ1+δ2+1){(δ1 t+δ2 t)(δ1 t+δ2 t+1)+1}={Tr(g)}−1,
→(δ1 2+δ2 2+δ1+δ2+1)[{(δ1+δ2)(δ1+δ2+1)}t+1]={Tr(g)}−1,
→(δ1 2+δ2 2+δ1+δ2+1){(δ1 2+δ2 2+δ1+δ2)t+1}={Tr(g)}−1,
→(δ1 2+δ2 2+δ1+δ2+1)(δ1 2+δ2 2+δ1+δ2+1)t ={Tr(g)}−1,
→(δ1 2+δ2 2+δ1+δ2+1)t+1 ={Tr(g)}−1,
→δ1 2+δ2 2+δ1+δ2+1=[{Tr(g)}−1]t−1,
→δ1 2+δ2 2+δ1+δ2+1={Tr(g)}1−t,
→δ1 2+δ2 2+δ1+δ2+1={Tr(g)}q−t (112)
X 2 +X+1={Tr(g)}q−t (113)
δ2 2+δ2 =X 2+(X 2 +X+1)X t(=X 2 +{Tr(g)}q−t X t) (114)
-
- Input: (δ1, δ2)
- Output: Tr(g), (b0, b1) (b0, b1 are additional bits)
-
- Input: Tr(g), (b0, b1)
- Output: (δ1, δ2)
Claims (5)
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JP5554357B2 (en) | 2012-03-15 | 2014-07-23 | 株式会社東芝 | Arithmetic unit |
Citations (14)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US20090207999A1 (en) | 2008-02-18 | 2009-08-20 | Kabushiki Kaisha Toshiba | Decryption processing apparatus, system, method, and computer program product |
US20100046743A1 (en) * | 2008-08-25 | 2010-02-25 | Kabushiki Kaisha Toshiba | Apparatus for performing data compression processing using algebraic torus |
US20100049777A1 (en) | 2008-08-25 | 2010-02-25 | Kabushiki Kaisha Toshiba | Representation converting apparatus, arithmetic apparatus, representation converting method, and computer program product |
US20100046746A1 (en) * | 2008-08-25 | 2010-02-25 | Kabushiki Kaisha Toshiba | Parameter generating device and cryptographic processing system |
US20100046745A1 (en) | 2008-08-25 | 2010-02-25 | Kabushiki Kaisha Toshiba | Encrypting apparatus, decrypting apparatus, cryptocommunication system, and methods and computer program products therefor |
US20100046742A1 (en) * | 2008-08-25 | 2010-02-25 | Kabushiki Kaisha Toshiba | Apparatus and computer program product for performing data compression processing using algebraic torus |
US20100046741A1 (en) * | 2008-08-25 | 2010-02-25 | Kabushiki Kaisha Toshiba | Apparatus, method, and computer program product for decrypting, and apparatus, method, and computer program product for encrypting |
US20100063986A1 (en) | 2008-09-10 | 2010-03-11 | Kabushiki Kaisha Toshiba | Computing device, method, and computer program product |
US20100226496A1 (en) * | 2009-03-04 | 2010-09-09 | Koichiro Akiyama | Encryption apparatus, decryption apparatus, key generation apparatus, and storage medium |
WO2010145983A1 (en) | 2009-06-16 | 2010-12-23 | Thomson Licensing | A method and a device for performing torus-based cryptography |
WO2011010383A1 (en) | 2009-07-23 | 2011-01-27 | 株式会社東芝 | Arithmetic device |
WO2011030468A1 (en) | 2009-09-14 | 2011-03-17 | 株式会社東芝 | Arithmetic device |
WO2011033672A1 (en) | 2009-09-18 | 2011-03-24 | 株式会社東芝 | Arithmetic apparatus, method and program |
US20130246489A1 (en) * | 2012-03-15 | 2013-09-19 | Tomoko Yonemura | Computing device |
-
2010
- 2010-12-09 JP JP2010275160A patent/JP5178810B2/en not_active Expired - Fee Related
-
2011
- 2011-09-06 US US13/225,964 patent/US8934631B2/en not_active Expired - Fee Related
Patent Citations (21)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US20090207999A1 (en) | 2008-02-18 | 2009-08-20 | Kabushiki Kaisha Toshiba | Decryption processing apparatus, system, method, and computer program product |
US8233616B2 (en) * | 2008-08-25 | 2012-07-31 | Kabushiki Kaisha Toshiba | Apparatus and computer program product for performing data compression processing using algebraic torus |
US20100046743A1 (en) * | 2008-08-25 | 2010-02-25 | Kabushiki Kaisha Toshiba | Apparatus for performing data compression processing using algebraic torus |
US20100049777A1 (en) | 2008-08-25 | 2010-02-25 | Kabushiki Kaisha Toshiba | Representation converting apparatus, arithmetic apparatus, representation converting method, and computer program product |
US20100046746A1 (en) * | 2008-08-25 | 2010-02-25 | Kabushiki Kaisha Toshiba | Parameter generating device and cryptographic processing system |
US20100046745A1 (en) | 2008-08-25 | 2010-02-25 | Kabushiki Kaisha Toshiba | Encrypting apparatus, decrypting apparatus, cryptocommunication system, and methods and computer program products therefor |
US20100046742A1 (en) * | 2008-08-25 | 2010-02-25 | Kabushiki Kaisha Toshiba | Apparatus and computer program product for performing data compression processing using algebraic torus |
US20100046741A1 (en) * | 2008-08-25 | 2010-02-25 | Kabushiki Kaisha Toshiba | Apparatus, method, and computer program product for decrypting, and apparatus, method, and computer program product for encrypting |
US8675874B2 (en) * | 2008-08-25 | 2014-03-18 | Kabushiki Kaisha Toshiba | Apparatus for performing data compression processing using algebraic torus |
US8533243B2 (en) * | 2008-08-25 | 2013-09-10 | Kabushiki Kaisha Toshiba | Representation converting apparatus, arithmetic apparatus, representation converting method, and computer program product |
US8438205B2 (en) * | 2008-09-10 | 2013-05-07 | Kabushiki Kaisha Toshiba | Exponentiation calculation apparatus and method for calculating square root in finite extension field |
US8543630B2 (en) * | 2008-09-10 | 2013-09-24 | Kabushiki Kaisha Toshiba | Exponentiation calculation apparatus and method for calculating square root in finite extension field |
US20100063986A1 (en) | 2008-09-10 | 2010-03-11 | Kabushiki Kaisha Toshiba | Computing device, method, and computer program product |
US8311215B2 (en) * | 2009-03-04 | 2012-11-13 | Kabushiki Kaisha Toshiba | Encryption apparatus, decryption apparatus, key generation apparatus, and storage medium |
US20100226496A1 (en) * | 2009-03-04 | 2010-09-09 | Koichiro Akiyama | Encryption apparatus, decryption apparatus, key generation apparatus, and storage medium |
WO2010145983A1 (en) | 2009-06-16 | 2010-12-23 | Thomson Licensing | A method and a device for performing torus-based cryptography |
US8548162B2 (en) * | 2009-06-16 | 2013-10-01 | Thomson Licensing | Method and a device for performing torus-based cryptography |
WO2011010383A1 (en) | 2009-07-23 | 2011-01-27 | 株式会社東芝 | Arithmetic device |
WO2011030468A1 (en) | 2009-09-14 | 2011-03-17 | 株式会社東芝 | Arithmetic device |
WO2011033672A1 (en) | 2009-09-18 | 2011-03-24 | 株式会社東芝 | Arithmetic apparatus, method and program |
US20130246489A1 (en) * | 2012-03-15 | 2013-09-19 | Tomoko Yonemura | Computing device |
Non-Patent Citations (5)
Title |
---|
K. Rubin and A. Silverberg, compression in finite fileds and torus-based cryptography, 2000, Mathematic subject classification, p. 1-28. * |
Karl Rubin, et al., "Torus-Based Cryptography," CRYPTO 2003, LNCS 2729, 2003, 17 pages. |
Koray Karabina, "Torus-Based Compression by Factor 4 and 6," Cryptology ePrint Archive, Report 2010/525, 20 pages. |
Marc Joye, On Cryptographic Schemes Based on Discrete Logarithms and Factoring, Lecture Notes in Computer Science, 2009, vol. 5888, pp. 41-52 with cover page. |
Office Action issued Oct. 2, 2012 in Japanese Patent Application No. 2010-275160 with English language translation. |
Cited By (2)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
US20140270159A1 (en) * | 2013-03-18 | 2014-09-18 | Electronics And Telecommunications Research Institute | System and method for providing compressed encryption and decryption in homomorphic encryption based on integers |
US9374220B2 (en) * | 2013-03-18 | 2016-06-21 | Electronics And Telecommunications Research Institute | System and method for providing compressed encryption and decryption in homomorphic encryption based on integers |
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US20120150931A1 (en) | 2012-06-14 |
JP5178810B2 (en) | 2013-04-10 |
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